This document describes research into using three disc-shaped robots to capture a convex object in the plane. It presents the problem, assumptions made, and provides examples of initial configurations where the object can escape and where it is immobilized. An algorithm is described to compute the "capture region" - all positions for the third robot that prevent the object from escaping. Both exact and approximate algorithms are presented and directions for future work are discussed.

1. Capturing a Convex Object with Three Discs
Jeff Erickson Shripad Thite Fred Rothganger Jean Ponce
jeffe, thite, rothgang, jponce @uiuc.edu
Department of Computer Science
University of Illinois at Urbana-Champaign
Urbana, Illinois 61801, USA
http://www-cvr.ai.uiuc.edu
September 17, 2003 @ ICRA (Taipei, Taiwan)

2. 2
Problem
b
a
P
Input:
– an arbitrary convex object in the plane
– three disc-shaped robots— , , and
Given:
– fixed positions of robots and
– position of in contact with and in its initial orientation
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

3. 3
Problem
b
a
P
c
Question: Where should robot be placed to capture ?
Capture region the set of all positions (if any) of that
prevent from escaping
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

4. 4
Problem
can perform arbitrary rigid motions
The interior of cannot intersect any robot
Capture prevent from escaping to infinity
Capture grasp (immobilize)
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

5. 5
Example: Initial confi guration
b
a
P
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

6. 6
Example: Escape through
b
a
P
c
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

7. 7
Example: Immobilized
b
a
P
c
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

8. 8
Example: Turning Counter-Clockwise
b
a
P
c
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

9. 9
Example: Escape through
b
a
P
c
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

10. 10
No Escape
a b
P
c
can spin around without being able to escape
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

11. 11
Motivation
http://www-cvr.ai.uiuc.edu/ponce grp/demo/robots/
Robots may be mobile platforms, fingertips of a gripper, pins
of a parts feeder.
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

12. 12
Motivation
Sensorless manipulation:
in-hand manipulation
mobile robot motion planning
regrasping [Sundsang, Phoka: ICRA ’03, Session WP9]
Some advantages:
contact need not be maintained during manipulation
no reliance on any particular model of friction
tolerance of uncertainty
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

13. 13
Related Work
Fixturing: [Gopal, Goldberg et al., ICRA 2002]
(quality of grasp = 1 / amount of wiggle room)
Manipulating algebraic parts in the plane [Rao,
Goldberg—ITRA ’95]
Graspless manipulation [Lynch—ICRA ’97; Lynch ’96]
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

14. 14
Previous Work
Maximum Independent Capture Discs (MICaDs)
[Sudsang, Ponce—ICRA ’98 and IEEE Trans. Rob. Autom.
2002]:
associate each robot with a fixed edge of the object
(the object is a triangle);
starting from an immobilizing configuration, compute
the maximum distance that one robot can travel while
keeping object captured
compute three maximum discs representing triples of
capturing configurations
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

15. 15
Previous Work
Capturing a concave polygon with two robots [Sudsang,
Luewirawong—ICRA 2003]
Caging planar bodies with one-parameter two-fingered
systems [Rimon, Blake—IJRR 2000]
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

16. 16
Some Assumptions
b
a
P
c
Without loss of generality
robot is at the origin
robot is on the positive -axis
, , and are labeled in clockwise order
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

17. 17
Simplifi cations for this Talk
b
a
P
c
is a polygon
Robots have zero radius (points)
All definitions still apply when and the robots are arbitrary
convex objects.
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

18. 18
Canonical Motion of
b
a
c
turns (counter)clockwise keeping in contact with and .
One of three things happens
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

19. 19
Canonical Motion of
b
a
b
a
c
c
(i) achieves an escape angle in which it escapes by pure translation
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

20. 20
Canonical Motion of
b
a
b
a
c
c
(ii) is blocked from turning further by (a triple contact)
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

21. 21
Canonical Motion of
a b
P
c
(iii) turns all the way around without being able to escape
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

22. 22
Main Lemma
Lemma: If can escape by any rigid motion, then can
escape by canonical motion.
Contra-positive: If cannot escape by canonical motion,
then cannot escape at all.
If can escape through , then in a preprocessing step we
compute the first escape angle such that in orientation
can escape through .
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

23. 23
Capture Region
the set of all positions of that prevent from escaping
We will concentrate on CCW-escape and computing
henceforth.
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

24. 24
Confi guration Space
is the position of and
is the orientation of
Without loss of generality, the initial orientation is .
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

25. 25
A -Slice of Confi guration Space
B*
A* θ
θ
b
a
Pθ
c
,
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

26. 26
Preventing Escape by Pure Translation
B*
A* θ
θ
b
a
Pθ
c
in orientation can escape by pure translation in one of
three ways:
Escape through : if int (equiv., int )
Escape through : if int
Escape through : if int
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

27. 27
A View of Confi guration Space
B*
A*
P*
+θ
, ,
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

28. 28
Visibility Problem
Consider a fixed position of robot
Let be the ray with origin at pointing in the -direction
(upwards).
What does intersect first?
Let be the value of where the first intersection of
occurs.
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

29. 29
Visibility Problem
If , then is not captured; otherwise, intersects one
of the following:
case : blocks in orientation from turning further,
i.e., is captured!
case : int and can escape through
case : int and can escape through
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

30. 30
Exact Algorithm
is the projection of the lower envelope of , , and .
is the projection of the upper envelope of , , and .
We can compute the capture region in
polynomial time.
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

31. 31
Approximate Algorithm
Compute the capture region restricted to a grid (a discrete
approximation with any desired accuracy).
Compute polyhedral approximations of , , and .
Render each with a -buffer using orientation as the depth.
The grid points where is visible constitute the
approximate capture region (and similarly for ).
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

32. 32
Approximate Capture Region: , ,
a b
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

33. 33
Approximate Capture Region: , ,
a b
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

34. 34
Approximate Capture Region: , ,
a b
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

35. 35
Conclusion
We characterized the capture region of a three robot that
capture a convex object in the plane in conjunction with two
fixed robots.
Reduced to a visibility problem in configuration space .
Computed the exact capture region in polynomial time.
Used graphics hardware ( -buffer) to compute a discrete
approximation quickly.
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

36. 36
Directions for Future Work
Capturing non-convex objects or linkages of convex objects
(canonical motion? two robots suffice?) [Sudsang,
Luewirawong—ICRA 2003]
Integrating the computation of the capture region into
algorithms for grasping, manipulation, and motion planning
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

37. 37
Thanks
Steve LaValle, Svetlana Lazebnik, and others
NSF grants IRI-9907009, CCR-0093348, and CCR-0219594
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

38. Appendix A:
Exact Algorithm

39. Appendix A: Exact Algorithm 39
Exact Algorithm
The boundary of is a piecewise-smooth collection of
algebraic surface patches of constant degree with no
self-intersections.
The number of patches (number of critical orientations) is
.
The critical orientations can be computed in time
by a rotating calipers algorithm.
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

40. Appendix A: Exact Algorithm 40
Exact Algorithm
Project boundary curves and silhouette curves of each of the
surface patches on to the plane.
Compute the arrangement of these curves with
cells.
The capture region is the union of cells in this cell
decomposition.
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

41. Appendix A: Exact Algorithm 41
Exact Algorithm
Compute a cylindrical decomposition of the volume between
the lower envelope of the patches above and the upper
envelope of patches below .
The intersection of each cylinder is the union of cells in the
cell decomposition—all these cells are marked once if the
object immediately above is either and marked again if
the object immediately below is .
The capture region is the union of twice-marked cells.
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

42. Appendix B:
First-Order Configuration Space

43. Appendix B: First-Order Configuration Space 43
Confi guration Space
is the position and is the orientation of
Without loss of generality, the initial orientation is .
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

44. Appendix B: First-Order Configuration Space 44
First-Order Obstacles
Fix the position of .
A B
p
C
, ,
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

45. Appendix B: First-Order Configuration Space 45
Escape by Pure Translation
in orientation can escape by pure translation in one of
three ways:
Escape through : if
Escape through : if
Escape through : if
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

46. Appendix B: First-Order Configuration Space 46
Escape by Pure Translation
b
a
b
b a
a
b b
a a
c
c
c
c
c
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

47. Appendix B: First-Order Configuration Space 47
Valid Confi gurations
, ,
Invalid region = int int int
Space of valid configurations
int and int and int
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

48. Appendix B: First-Order Configuration Space 48
Structure of Valid Space
Every rigid motion of that does not intersect any robot
corresponds to a path in valid space.
is captured if and only if the component of valid space
containing the initial configuration is compact (closed and
bounded).
If can escape, then there exists a such that in
orientation can escape by pure translation.
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

49. Appendix B: First-Order Configuration Space 49
Escape Orientation
Necessary and sufficient condition for escape: some pair of
obstacles do not intersect, i.e., either
1. , or
2. , or
3.
(Counterclockwise) Escape angle =
, or , or
(If CCW escape is not possible, then .)
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

50. Appendix B: First-Order Configuration Space 50
Canonical Motion Revisited
Canonical motion of is motion along .
Triple contact:
Initial configuration is valid, i.e., .
Counterclockwise escape angle:
, or , or
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

51. Appendix B: First-Order Configuration Space 51
Relation between Two Confi guration Spaces
A B
B*
A* θ
θ
b
a
p
Pθ
C c
(equiv., )
Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu

52. The End